Theorem omp1eomlem 7123

Description: Lemma for omp1eom 7124. (Contributed by Jim Kingdon, 11-Jul-2023.)

Hypotheses

Ref	Expression
omp1eom.f	⊢ 𝐹 = (𝑥 ∈ ω ↦ if(𝑥 = ∅, (inr‘𝑥), (inl‘∪ 𝑥)))
omp1eom.s	⊢ 𝑆 = (𝑥 ∈ ω ↦ suc 𝑥)
omp1eom.g	⊢ 𝐺 = case(𝑆, ( I ↾ 1o))

Assertion

Ref	Expression
omp1eomlem	⊢ 𝐹:ω–1-1-onto→(ω ⊔ 1o)

Distinct variable group:   𝑥,𝐺

Allowed substitution hints:   𝑆(𝑥)   𝐹(𝑥)

Proof of Theorem omp1eomlem

Dummy variables 𝑧 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		omp1eom.f	. . 3 ⊢ 𝐹 = (𝑥 ∈ ω ↦ if(𝑥 = ∅, (inr‘𝑥), (inl‘∪ 𝑥)))
2		el1o 6462	. . . . . . 7 ⊢ (𝑥 ∈ 1o ↔ 𝑥 = ∅)
3	2	biimpri 133	. . . . . 6 ⊢ (𝑥 = ∅ → 𝑥 ∈ 1o)
4	3	adantl 277	. . . . 5 ⊢ (((⊤ ∧ 𝑥 ∈ ω) ∧ 𝑥 = ∅) → 𝑥 ∈ 1o)
5		djurcl 7081	. . . . 5 ⊢ (𝑥 ∈ 1o → (inr‘𝑥) ∈ (ω ⊔ 1o))
6	4, 5	syl 14	. . . 4 ⊢ (((⊤ ∧ 𝑥 ∈ ω) ∧ 𝑥 = ∅) → (inr‘𝑥) ∈ (ω ⊔ 1o))
7		nnpredcl 4640	. . . . . 6 ⊢ (𝑥 ∈ ω → ∪ 𝑥 ∈ ω)
8	7	ad2antlr 489	. . . . 5 ⊢ (((⊤ ∧ 𝑥 ∈ ω) ∧ ¬ 𝑥 = ∅) → ∪ 𝑥 ∈ ω)
9		djulcl 7080	. . . . 5 ⊢ (∪ 𝑥 ∈ ω → (inl‘∪ 𝑥) ∈ (ω ⊔ 1o))
10	8, 9	syl 14	. . . 4 ⊢ (((⊤ ∧ 𝑥 ∈ ω) ∧ ¬ 𝑥 = ∅) → (inl‘∪ 𝑥) ∈ (ω ⊔ 1o))
11		nndceq0 4635	. . . . 5 ⊢ (𝑥 ∈ ω → DECID 𝑥 = ∅)
12	11	adantl 277	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ω) → DECID 𝑥 = ∅)
13	6, 10, 12	ifcldadc 3578	. . 3 ⊢ ((⊤ ∧ 𝑥 ∈ ω) → if(𝑥 = ∅, (inr‘𝑥), (inl‘∪ 𝑥)) ∈ (ω ⊔ 1o))
14		omp1eom.s	. . . . . . . 8 ⊢ 𝑆 = (𝑥 ∈ ω ↦ suc 𝑥)
15		peano2 4612	. . . . . . . 8 ⊢ (𝑥 ∈ ω → suc 𝑥 ∈ ω)
16	14, 15	fmpti 5689	. . . . . . 7 ⊢ 𝑆:ω⟶ω
17	16	a1i 9	. . . . . 6 ⊢ (⊤ → 𝑆:ω⟶ω)
18		f1oi 5518	. . . . . . . . 9 ⊢ ( I ↾ 1o):1o–1-1-onto→1o
19		f1of 5480	. . . . . . . . 9 ⊢ (( I ↾ 1o):1o–1-1-onto→1o → ( I ↾ 1o):1o⟶1o)
20	18, 19	ax-mp 5	. . . . . . . 8 ⊢ ( I ↾ 1o):1o⟶1o
21		1onn 6545	. . . . . . . . 9 ⊢ 1o ∈ ω
22		omelon 4626	. . . . . . . . . 10 ⊢ ω ∈ On
23	22	onelssi 4447	. . . . . . . . 9 ⊢ (1o ∈ ω → 1o ⊆ ω)
24	21, 23	ax-mp 5	. . . . . . . 8 ⊢ 1o ⊆ ω
25		fss 5396	. . . . . . . 8 ⊢ ((( I ↾ 1o):1o⟶1o ∧ 1o ⊆ ω) → ( I ↾ 1o):1o⟶ω)
26	20, 24, 25	mp2an 426	. . . . . . 7 ⊢ ( I ↾ 1o):1o⟶ω
27	26	a1i 9	. . . . . 6 ⊢ (⊤ → ( I ↾ 1o):1o⟶ω)
28	17, 27	casef 7117	. . . . 5 ⊢ (⊤ → case(𝑆, ( I ↾ 1o)):(ω ⊔ 1o)⟶ω)
29		omp1eom.g	. . . . . 6 ⊢ 𝐺 = case(𝑆, ( I ↾ 1o))
30	29	feq1i 5377	. . . . 5 ⊢ (𝐺:(ω ⊔ 1o)⟶ω ↔ case(𝑆, ( I ↾ 1o)):(ω ⊔ 1o)⟶ω)
31	28, 30	sylibr 134	. . . 4 ⊢ (⊤ → 𝐺:(ω ⊔ 1o)⟶ω)
32	31	ffvelcdmda 5672	. . 3 ⊢ ((⊤ ∧ 𝑦 ∈ (ω ⊔ 1o)) → (𝐺‘𝑦) ∈ ω)
33		ffn 5384	. . . . . . . . . . . . . . . 16 ⊢ (𝑆:ω⟶ω → 𝑆 Fn ω)
34	16, 33	mp1i 10	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ ω ∧ 𝑦 = (inl‘𝑧)) → 𝑆 Fn ω)
35		ffun 5387	. . . . . . . . . . . . . . . 16 ⊢ (( I ↾ 1o):1o⟶1o → Fun ( I ↾ 1o))
36	20, 35	mp1i 10	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ ω ∧ 𝑦 = (inl‘𝑧)) → Fun ( I ↾ 1o))
37		simpl 109	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ ω ∧ 𝑦 = (inl‘𝑧)) → 𝑧 ∈ ω)
38	34, 36, 37	caseinl 7120	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ ω ∧ 𝑦 = (inl‘𝑧)) → (case(𝑆, ( I ↾ 1o))‘(inl‘𝑧)) = (𝑆‘𝑧))
39	29	eqcomi 2193	. . . . . . . . . . . . . . . 16 ⊢ case(𝑆, ( I ↾ 1o)) = 𝐺
40	39	a1i 9	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ ω ∧ 𝑦 = (inl‘𝑧)) → case(𝑆, ( I ↾ 1o)) = 𝐺)
41		simpr 110	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ ω ∧ 𝑦 = (inl‘𝑧)) → 𝑦 = (inl‘𝑧))
42	41	eqcomd 2195	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ ω ∧ 𝑦 = (inl‘𝑧)) → (inl‘𝑧) = 𝑦)
43	40, 42	fveq12d 5541	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ ω ∧ 𝑦 = (inl‘𝑧)) → (case(𝑆, ( I ↾ 1o))‘(inl‘𝑧)) = (𝐺‘𝑦))
44		peano2 4612	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ ω → suc 𝑧 ∈ ω)
45		suceq 4420	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑧 → suc 𝑥 = suc 𝑧)
46	45, 14	fvmptg 5613	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ ω ∧ suc 𝑧 ∈ ω) → (𝑆‘𝑧) = suc 𝑧)
47	44, 46	mpdan 421	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ ω → (𝑆‘𝑧) = suc 𝑧)
48	47	adantr 276	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ ω ∧ 𝑦 = (inl‘𝑧)) → (𝑆‘𝑧) = suc 𝑧)
49	38, 43, 48	3eqtr3d 2230	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ ω ∧ 𝑦 = (inl‘𝑧)) → (𝐺‘𝑦) = suc 𝑧)
50		peano3 4613	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ ω → suc 𝑧 ≠ ∅)
51	50	adantr 276	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ ω ∧ 𝑦 = (inl‘𝑧)) → suc 𝑧 ≠ ∅)
52	49, 51	eqnetrd 2384	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ω ∧ 𝑦 = (inl‘𝑧)) → (𝐺‘𝑦) ≠ ∅)
53	52	adantl 277	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ ω ∧ 𝑦 ∈ (ω ⊔ 1o)) ∧ 𝑥 = ∅) ∧ (𝑧 ∈ ω ∧ 𝑦 = (inl‘𝑧))) → (𝐺‘𝑦) ≠ ∅)
54	53	necomd 2446	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ ω ∧ 𝑦 ∈ (ω ⊔ 1o)) ∧ 𝑥 = ∅) ∧ (𝑧 ∈ ω ∧ 𝑦 = (inl‘𝑧))) → ∅ ≠ (𝐺‘𝑦))
55	54	neneqd 2381	. . . . . . . . 9 ⊢ ((((𝑥 ∈ ω ∧ 𝑦 ∈ (ω ⊔ 1o)) ∧ 𝑥 = ∅) ∧ (𝑧 ∈ ω ∧ 𝑦 = (inl‘𝑧))) → ¬ ∅ = (𝐺‘𝑦))
56		simplr 528	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ ω ∧ 𝑦 ∈ (ω ⊔ 1o)) ∧ 𝑥 = ∅) ∧ (𝑧 ∈ ω ∧ 𝑦 = (inl‘𝑧))) → 𝑥 = ∅)
57	56	eqeq1d 2198	. . . . . . . . 9 ⊢ ((((𝑥 ∈ ω ∧ 𝑦 ∈ (ω ⊔ 1o)) ∧ 𝑥 = ∅) ∧ (𝑧 ∈ ω ∧ 𝑦 = (inl‘𝑧))) → (𝑥 = (𝐺‘𝑦) ↔ ∅ = (𝐺‘𝑦)))
58	55, 57	mtbird 674	. . . . . . . 8 ⊢ ((((𝑥 ∈ ω ∧ 𝑦 ∈ (ω ⊔ 1o)) ∧ 𝑥 = ∅) ∧ (𝑧 ∈ ω ∧ 𝑦 = (inl‘𝑧))) → ¬ 𝑥 = (𝐺‘𝑦))
59		djune 7107	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ V ∧ 𝑥 ∈ V) → (inl‘𝑧) ≠ (inr‘𝑥))
60	59	elvd 2757	. . . . . . . . . . 11 ⊢ (𝑧 ∈ V → (inl‘𝑧) ≠ (inr‘𝑥))
61	60	elv 2756	. . . . . . . . . 10 ⊢ (inl‘𝑧) ≠ (inr‘𝑥)
62	61	neii 2362	. . . . . . . . 9 ⊢ ¬ (inl‘𝑧) = (inr‘𝑥)
63		simprr 531	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ ω ∧ 𝑦 ∈ (ω ⊔ 1o)) ∧ 𝑥 = ∅) ∧ (𝑧 ∈ ω ∧ 𝑦 = (inl‘𝑧))) → 𝑦 = (inl‘𝑧))
64		simpr 110	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ (ω ⊔ 1o)) ∧ 𝑥 = ∅) → 𝑥 = ∅)
65	64	iftrued 3556	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ (ω ⊔ 1o)) ∧ 𝑥 = ∅) → if(𝑥 = ∅, (inr‘𝑥), (inl‘∪ 𝑥)) = (inr‘𝑥))
66	65	adantr 276	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ ω ∧ 𝑦 ∈ (ω ⊔ 1o)) ∧ 𝑥 = ∅) ∧ (𝑧 ∈ ω ∧ 𝑦 = (inl‘𝑧))) → if(𝑥 = ∅, (inr‘𝑥), (inl‘∪ 𝑥)) = (inr‘𝑥))
67	63, 66	eqeq12d 2204	. . . . . . . . 9 ⊢ ((((𝑥 ∈ ω ∧ 𝑦 ∈ (ω ⊔ 1o)) ∧ 𝑥 = ∅) ∧ (𝑧 ∈ ω ∧ 𝑦 = (inl‘𝑧))) → (𝑦 = if(𝑥 = ∅, (inr‘𝑥), (inl‘∪ 𝑥)) ↔ (inl‘𝑧) = (inr‘𝑥)))
68	62, 67	mtbiri 676	. . . . . . . 8 ⊢ ((((𝑥 ∈ ω ∧ 𝑦 ∈ (ω ⊔ 1o)) ∧ 𝑥 = ∅) ∧ (𝑧 ∈ ω ∧ 𝑦 = (inl‘𝑧))) → ¬ 𝑦 = if(𝑥 = ∅, (inr‘𝑥), (inl‘∪ 𝑥)))
69	58, 68	2falsed 703	. . . . . . 7 ⊢ ((((𝑥 ∈ ω ∧ 𝑦 ∈ (ω ⊔ 1o)) ∧ 𝑥 = ∅) ∧ (𝑧 ∈ ω ∧ 𝑦 = (inl‘𝑧))) → (𝑥 = (𝐺‘𝑦) ↔ 𝑦 = if(𝑥 = ∅, (inr‘𝑥), (inl‘∪ 𝑥))))
70	69	rexlimdvaa 2608	. . . . . 6 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ (ω ⊔ 1o)) ∧ 𝑥 = ∅) → (∃𝑧 ∈ ω 𝑦 = (inl‘𝑧) → (𝑥 = (𝐺‘𝑦) ↔ 𝑦 = if(𝑥 = ∅, (inr‘𝑥), (inl‘∪ 𝑥)))))
71		simplr 528	. . . . . . . . 9 ⊢ ((((𝑥 ∈ ω ∧ 𝑦 ∈ (ω ⊔ 1o)) ∧ 𝑥 = ∅) ∧ (𝑧 ∈ 1o ∧ 𝑦 = (inr‘𝑧))) → 𝑥 = ∅)
72	29	a1i 9	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ 1o ∧ 𝑦 = (inr‘𝑧)) → 𝐺 = case(𝑆, ( I ↾ 1o)))
73		simpr 110	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ 1o ∧ 𝑦 = (inr‘𝑧)) → 𝑦 = (inr‘𝑧))
74	72, 73	fveq12d 5541	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ 1o ∧ 𝑦 = (inr‘𝑧)) → (𝐺‘𝑦) = (case(𝑆, ( I ↾ 1o))‘(inr‘𝑧)))
75	14	funmpt2 5274	. . . . . . . . . . . . . 14 ⊢ Fun 𝑆
76	75	a1i 9	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ 1o ∧ 𝑦 = (inr‘𝑧)) → Fun 𝑆)
77		fnresi 5352	. . . . . . . . . . . . . 14 ⊢ ( I ↾ 1o) Fn 1o
78	77	a1i 9	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ 1o ∧ 𝑦 = (inr‘𝑧)) → ( I ↾ 1o) Fn 1o)
79		simpl 109	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ 1o ∧ 𝑦 = (inr‘𝑧)) → 𝑧 ∈ 1o)
80	76, 78, 79	caseinr 7121	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ 1o ∧ 𝑦 = (inr‘𝑧)) → (case(𝑆, ( I ↾ 1o))‘(inr‘𝑧)) = (( I ↾ 1o)‘𝑧))
81		fvresi 5730	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 1o → (( I ↾ 1o)‘𝑧) = 𝑧)
82	81	adantr 276	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ 1o ∧ 𝑦 = (inr‘𝑧)) → (( I ↾ 1o)‘𝑧) = 𝑧)
83	80, 82	eqtrd 2222	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ 1o ∧ 𝑦 = (inr‘𝑧)) → (case(𝑆, ( I ↾ 1o))‘(inr‘𝑧)) = 𝑧)
84		el1o 6462	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ 1o ↔ 𝑧 = ∅)
85	79, 84	sylib 122	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ 1o ∧ 𝑦 = (inr‘𝑧)) → 𝑧 = ∅)
86	74, 83, 85	3eqtrd 2226	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 1o ∧ 𝑦 = (inr‘𝑧)) → (𝐺‘𝑦) = ∅)
87	86	adantl 277	. . . . . . . . 9 ⊢ ((((𝑥 ∈ ω ∧ 𝑦 ∈ (ω ⊔ 1o)) ∧ 𝑥 = ∅) ∧ (𝑧 ∈ 1o ∧ 𝑦 = (inr‘𝑧))) → (𝐺‘𝑦) = ∅)
88	71, 87	eqtr4d 2225	. . . . . . . 8 ⊢ ((((𝑥 ∈ ω ∧ 𝑦 ∈ (ω ⊔ 1o)) ∧ 𝑥 = ∅) ∧ (𝑧 ∈ 1o ∧ 𝑦 = (inr‘𝑧))) → 𝑥 = (𝐺‘𝑦))
89	85	adantl 277	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ ω ∧ 𝑦 ∈ (ω ⊔ 1o)) ∧ 𝑥 = ∅) ∧ (𝑧 ∈ 1o ∧ 𝑦 = (inr‘𝑧))) → 𝑧 = ∅)
90	71, 89	eqtr4d 2225	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ ω ∧ 𝑦 ∈ (ω ⊔ 1o)) ∧ 𝑥 = ∅) ∧ (𝑧 ∈ 1o ∧ 𝑦 = (inr‘𝑧))) → 𝑥 = 𝑧)
91	90	fveq2d 5538	. . . . . . . . 9 ⊢ ((((𝑥 ∈ ω ∧ 𝑦 ∈ (ω ⊔ 1o)) ∧ 𝑥 = ∅) ∧ (𝑧 ∈ 1o ∧ 𝑦 = (inr‘𝑧))) → (inr‘𝑥) = (inr‘𝑧))
92	65	adantr 276	. . . . . . . . 9 ⊢ ((((𝑥 ∈ ω ∧ 𝑦 ∈ (ω ⊔ 1o)) ∧ 𝑥 = ∅) ∧ (𝑧 ∈ 1o ∧ 𝑦 = (inr‘𝑧))) → if(𝑥 = ∅, (inr‘𝑥), (inl‘∪ 𝑥)) = (inr‘𝑥))
93		simprr 531	. . . . . . . . 9 ⊢ ((((𝑥 ∈ ω ∧ 𝑦 ∈ (ω ⊔ 1o)) ∧ 𝑥 = ∅) ∧ (𝑧 ∈ 1o ∧ 𝑦 = (inr‘𝑧))) → 𝑦 = (inr‘𝑧))
94	91, 92, 93	3eqtr4rd 2233	. . . . . . . 8 ⊢ ((((𝑥 ∈ ω ∧ 𝑦 ∈ (ω ⊔ 1o)) ∧ 𝑥 = ∅) ∧ (𝑧 ∈ 1o ∧ 𝑦 = (inr‘𝑧))) → 𝑦 = if(𝑥 = ∅, (inr‘𝑥), (inl‘∪ 𝑥)))
95	88, 94	2thd 175	. . . . . . 7 ⊢ ((((𝑥 ∈ ω ∧ 𝑦 ∈ (ω ⊔ 1o)) ∧ 𝑥 = ∅) ∧ (𝑧 ∈ 1o ∧ 𝑦 = (inr‘𝑧))) → (𝑥 = (𝐺‘𝑦) ↔ 𝑦 = if(𝑥 = ∅, (inr‘𝑥), (inl‘∪ 𝑥))))
96	95	rexlimdvaa 2608	. . . . . 6 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ (ω ⊔ 1o)) ∧ 𝑥 = ∅) → (∃𝑧 ∈ 1o 𝑦 = (inr‘𝑧) → (𝑥 = (𝐺‘𝑦) ↔ 𝑦 = if(𝑥 = ∅, (inr‘𝑥), (inl‘∪ 𝑥)))))
97		djur 7098	. . . . . . . 8 ⊢ (𝑦 ∈ (ω ⊔ 1o) ↔ (∃𝑧 ∈ ω 𝑦 = (inl‘𝑧) ∨ ∃𝑧 ∈ 1o 𝑦 = (inr‘𝑧)))
98	97	biimpi 120	. . . . . . 7 ⊢ (𝑦 ∈ (ω ⊔ 1o) → (∃𝑧 ∈ ω 𝑦 = (inl‘𝑧) ∨ ∃𝑧 ∈ 1o 𝑦 = (inr‘𝑧)))
99	98	ad2antlr 489	. . . . . 6 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ (ω ⊔ 1o)) ∧ 𝑥 = ∅) → (∃𝑧 ∈ ω 𝑦 = (inl‘𝑧) ∨ ∃𝑧 ∈ 1o 𝑦 = (inr‘𝑧)))
100	70, 96, 99	mpjaod 719	. . . . 5 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ (ω ⊔ 1o)) ∧ 𝑥 = ∅) → (𝑥 = (𝐺‘𝑦) ↔ 𝑦 = if(𝑥 = ∅, (inr‘𝑥), (inl‘∪ 𝑥))))
101		simplll 533	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ ω ∧ 𝑦 ∈ (ω ⊔ 1o)) ∧ ¬ 𝑥 = ∅) ∧ (𝑧 ∈ ω ∧ 𝑦 = (inl‘𝑧))) → 𝑥 ∈ ω)
102		simplr 528	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ ω ∧ 𝑦 ∈ (ω ⊔ 1o)) ∧ ¬ 𝑥 = ∅) ∧ (𝑧 ∈ ω ∧ 𝑦 = (inl‘𝑧))) → ¬ 𝑥 = ∅)
103	102	neqned 2367	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ ω ∧ 𝑦 ∈ (ω ⊔ 1o)) ∧ ¬ 𝑥 = ∅) ∧ (𝑧 ∈ ω ∧ 𝑦 = (inl‘𝑧))) → 𝑥 ≠ ∅)
104		nnsucpred 4634	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ω ∧ 𝑥 ≠ ∅) → suc ∪ 𝑥 = 𝑥)
105	101, 103, 104	syl2anc 411	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ ω ∧ 𝑦 ∈ (ω ⊔ 1o)) ∧ ¬ 𝑥 = ∅) ∧ (𝑧 ∈ ω ∧ 𝑦 = (inl‘𝑧))) → suc ∪ 𝑥 = 𝑥)
106	105	eqeq2d 2201	. . . . . . . . 9 ⊢ ((((𝑥 ∈ ω ∧ 𝑦 ∈ (ω ⊔ 1o)) ∧ ¬ 𝑥 = ∅) ∧ (𝑧 ∈ ω ∧ 𝑦 = (inl‘𝑧))) → (suc 𝑧 = suc ∪ 𝑥 ↔ suc 𝑧 = 𝑥))
107		eqcom 2191	. . . . . . . . 9 ⊢ (suc 𝑧 = 𝑥 ↔ 𝑥 = suc 𝑧)
108	106, 107	bitrdi 196	. . . . . . . 8 ⊢ ((((𝑥 ∈ ω ∧ 𝑦 ∈ (ω ⊔ 1o)) ∧ ¬ 𝑥 = ∅) ∧ (𝑧 ∈ ω ∧ 𝑦 = (inl‘𝑧))) → (suc 𝑧 = suc ∪ 𝑥 ↔ 𝑥 = suc 𝑧))
109		simprr 531	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ ω ∧ 𝑦 ∈ (ω ⊔ 1o)) ∧ ¬ 𝑥 = ∅) ∧ (𝑧 ∈ ω ∧ 𝑦 = (inl‘𝑧))) → 𝑦 = (inl‘𝑧))
110		simpr 110	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ (ω ⊔ 1o)) ∧ ¬ 𝑥 = ∅) → ¬ 𝑥 = ∅)
111	110	iffalsed 3559	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ (ω ⊔ 1o)) ∧ ¬ 𝑥 = ∅) → if(𝑥 = ∅, (inr‘𝑥), (inl‘∪ 𝑥)) = (inl‘∪ 𝑥))
112	111	adantr 276	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ ω ∧ 𝑦 ∈ (ω ⊔ 1o)) ∧ ¬ 𝑥 = ∅) ∧ (𝑧 ∈ ω ∧ 𝑦 = (inl‘𝑧))) → if(𝑥 = ∅, (inr‘𝑥), (inl‘∪ 𝑥)) = (inl‘∪ 𝑥))
113	109, 112	eqeq12d 2204	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ ω ∧ 𝑦 ∈ (ω ⊔ 1o)) ∧ ¬ 𝑥 = ∅) ∧ (𝑧 ∈ ω ∧ 𝑦 = (inl‘𝑧))) → (𝑦 = if(𝑥 = ∅, (inr‘𝑥), (inl‘∪ 𝑥)) ↔ (inl‘𝑧) = (inl‘∪ 𝑥)))
114		vuniex 4456	. . . . . . . . . . . 12 ⊢ ∪ 𝑥 ∈ V
115		inl11 7094	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ V ∧ ∪ 𝑥 ∈ V) → ((inl‘𝑧) = (inl‘∪ 𝑥) ↔ 𝑧 = ∪ 𝑥))
116	114, 115	mpan2 425	. . . . . . . . . . 11 ⊢ (𝑧 ∈ V → ((inl‘𝑧) = (inl‘∪ 𝑥) ↔ 𝑧 = ∪ 𝑥))
117	116	elv 2756	. . . . . . . . . 10 ⊢ ((inl‘𝑧) = (inl‘∪ 𝑥) ↔ 𝑧 = ∪ 𝑥)
118	113, 117	bitrdi 196	. . . . . . . . 9 ⊢ ((((𝑥 ∈ ω ∧ 𝑦 ∈ (ω ⊔ 1o)) ∧ ¬ 𝑥 = ∅) ∧ (𝑧 ∈ ω ∧ 𝑦 = (inl‘𝑧))) → (𝑦 = if(𝑥 = ∅, (inr‘𝑥), (inl‘∪ 𝑥)) ↔ 𝑧 = ∪ 𝑥))
119		nnon 4627	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ω → 𝑧 ∈ On)
120	119	ad2antrl 490	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ ω ∧ 𝑦 ∈ (ω ⊔ 1o)) ∧ ¬ 𝑥 = ∅) ∧ (𝑧 ∈ ω ∧ 𝑦 = (inl‘𝑧))) → 𝑧 ∈ On)
121	7	ad3antrrr 492	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ ω ∧ 𝑦 ∈ (ω ⊔ 1o)) ∧ ¬ 𝑥 = ∅) ∧ (𝑧 ∈ ω ∧ 𝑦 = (inl‘𝑧))) → ∪ 𝑥 ∈ ω)
122		nnon 4627	. . . . . . . . . . 11 ⊢ (∪ 𝑥 ∈ ω → ∪ 𝑥 ∈ On)
123	121, 122	syl 14	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ ω ∧ 𝑦 ∈ (ω ⊔ 1o)) ∧ ¬ 𝑥 = ∅) ∧ (𝑧 ∈ ω ∧ 𝑦 = (inl‘𝑧))) → ∪ 𝑥 ∈ On)
124		suc11 4575	. . . . . . . . . 10 ⊢ ((𝑧 ∈ On ∧ ∪ 𝑥 ∈ On) → (suc 𝑧 = suc ∪ 𝑥 ↔ 𝑧 = ∪ 𝑥))
125	120, 123, 124	syl2anc 411	. . . . . . . . 9 ⊢ ((((𝑥 ∈ ω ∧ 𝑦 ∈ (ω ⊔ 1o)) ∧ ¬ 𝑥 = ∅) ∧ (𝑧 ∈ ω ∧ 𝑦 = (inl‘𝑧))) → (suc 𝑧 = suc ∪ 𝑥 ↔ 𝑧 = ∪ 𝑥))
126	118, 125	bitr4d 191	. . . . . . . 8 ⊢ ((((𝑥 ∈ ω ∧ 𝑦 ∈ (ω ⊔ 1o)) ∧ ¬ 𝑥 = ∅) ∧ (𝑧 ∈ ω ∧ 𝑦 = (inl‘𝑧))) → (𝑦 = if(𝑥 = ∅, (inr‘𝑥), (inl‘∪ 𝑥)) ↔ suc 𝑧 = suc ∪ 𝑥))
127	49	adantl 277	. . . . . . . . 9 ⊢ ((((𝑥 ∈ ω ∧ 𝑦 ∈ (ω ⊔ 1o)) ∧ ¬ 𝑥 = ∅) ∧ (𝑧 ∈ ω ∧ 𝑦 = (inl‘𝑧))) → (𝐺‘𝑦) = suc 𝑧)
128	127	eqeq2d 2201	. . . . . . . 8 ⊢ ((((𝑥 ∈ ω ∧ 𝑦 ∈ (ω ⊔ 1o)) ∧ ¬ 𝑥 = ∅) ∧ (𝑧 ∈ ω ∧ 𝑦 = (inl‘𝑧))) → (𝑥 = (𝐺‘𝑦) ↔ 𝑥 = suc 𝑧))
129	108, 126, 128	3bitr4rd 221	. . . . . . 7 ⊢ ((((𝑥 ∈ ω ∧ 𝑦 ∈ (ω ⊔ 1o)) ∧ ¬ 𝑥 = ∅) ∧ (𝑧 ∈ ω ∧ 𝑦 = (inl‘𝑧))) → (𝑥 = (𝐺‘𝑦) ↔ 𝑦 = if(𝑥 = ∅, (inr‘𝑥), (inl‘∪ 𝑥))))
130	129	rexlimdvaa 2608	. . . . . 6 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ (ω ⊔ 1o)) ∧ ¬ 𝑥 = ∅) → (∃𝑧 ∈ ω 𝑦 = (inl‘𝑧) → (𝑥 = (𝐺‘𝑦) ↔ 𝑦 = if(𝑥 = ∅, (inr‘𝑥), (inl‘∪ 𝑥)))))
131		simplr 528	. . . . . . . . 9 ⊢ ((((𝑥 ∈ ω ∧ 𝑦 ∈ (ω ⊔ 1o)) ∧ ¬ 𝑥 = ∅) ∧ (𝑧 ∈ 1o ∧ 𝑦 = (inr‘𝑧))) → ¬ 𝑥 = ∅)
132	86	adantl 277	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ ω ∧ 𝑦 ∈ (ω ⊔ 1o)) ∧ ¬ 𝑥 = ∅) ∧ (𝑧 ∈ 1o ∧ 𝑦 = (inr‘𝑧))) → (𝐺‘𝑦) = ∅)
133	132	eqeq2d 2201	. . . . . . . . 9 ⊢ ((((𝑥 ∈ ω ∧ 𝑦 ∈ (ω ⊔ 1o)) ∧ ¬ 𝑥 = ∅) ∧ (𝑧 ∈ 1o ∧ 𝑦 = (inr‘𝑧))) → (𝑥 = (𝐺‘𝑦) ↔ 𝑥 = ∅))
134	131, 133	mtbird 674	. . . . . . . 8 ⊢ ((((𝑥 ∈ ω ∧ 𝑦 ∈ (ω ⊔ 1o)) ∧ ¬ 𝑥 = ∅) ∧ (𝑧 ∈ 1o ∧ 𝑦 = (inr‘𝑧))) → ¬ 𝑥 = (𝐺‘𝑦))
135		djune 7107	. . . . . . . . . . . 12 ⊢ ((∪ 𝑥 ∈ V ∧ 𝑧 ∈ V) → (inl‘∪ 𝑥) ≠ (inr‘𝑧))
136	135	elvd 2757	. . . . . . . . . . 11 ⊢ (∪ 𝑥 ∈ V → (inl‘∪ 𝑥) ≠ (inr‘𝑧))
137	114, 136	ax-mp 5	. . . . . . . . . 10 ⊢ (inl‘∪ 𝑥) ≠ (inr‘𝑧)
138	137	nesymi 2406	. . . . . . . . 9 ⊢ ¬ (inr‘𝑧) = (inl‘∪ 𝑥)
139	73, 111	eqeqan12rd 2206	. . . . . . . . 9 ⊢ ((((𝑥 ∈ ω ∧ 𝑦 ∈ (ω ⊔ 1o)) ∧ ¬ 𝑥 = ∅) ∧ (𝑧 ∈ 1o ∧ 𝑦 = (inr‘𝑧))) → (𝑦 = if(𝑥 = ∅, (inr‘𝑥), (inl‘∪ 𝑥)) ↔ (inr‘𝑧) = (inl‘∪ 𝑥)))
140	138, 139	mtbiri 676	. . . . . . . 8 ⊢ ((((𝑥 ∈ ω ∧ 𝑦 ∈ (ω ⊔ 1o)) ∧ ¬ 𝑥 = ∅) ∧ (𝑧 ∈ 1o ∧ 𝑦 = (inr‘𝑧))) → ¬ 𝑦 = if(𝑥 = ∅, (inr‘𝑥), (inl‘∪ 𝑥)))
141	134, 140	2falsed 703	. . . . . . 7 ⊢ ((((𝑥 ∈ ω ∧ 𝑦 ∈ (ω ⊔ 1o)) ∧ ¬ 𝑥 = ∅) ∧ (𝑧 ∈ 1o ∧ 𝑦 = (inr‘𝑧))) → (𝑥 = (𝐺‘𝑦) ↔ 𝑦 = if(𝑥 = ∅, (inr‘𝑥), (inl‘∪ 𝑥))))
142	141	rexlimdvaa 2608	. . . . . 6 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ (ω ⊔ 1o)) ∧ ¬ 𝑥 = ∅) → (∃𝑧 ∈ 1o 𝑦 = (inr‘𝑧) → (𝑥 = (𝐺‘𝑦) ↔ 𝑦 = if(𝑥 = ∅, (inr‘𝑥), (inl‘∪ 𝑥)))))
143	98	ad2antlr 489	. . . . . 6 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ (ω ⊔ 1o)) ∧ ¬ 𝑥 = ∅) → (∃𝑧 ∈ ω 𝑦 = (inl‘𝑧) ∨ ∃𝑧 ∈ 1o 𝑦 = (inr‘𝑧)))
144	130, 142, 143	mpjaod 719	. . . . 5 ⊢ (((𝑥 ∈ ω ∧ 𝑦 ∈ (ω ⊔ 1o)) ∧ ¬ 𝑥 = ∅) → (𝑥 = (𝐺‘𝑦) ↔ 𝑦 = if(𝑥 = ∅, (inr‘𝑥), (inl‘∪ 𝑥))))
145		exmiddc 837	. . . . . . 7 ⊢ (DECID 𝑥 = ∅ → (𝑥 = ∅ ∨ ¬ 𝑥 = ∅))
146	11, 145	syl 14	. . . . . 6 ⊢ (𝑥 ∈ ω → (𝑥 = ∅ ∨ ¬ 𝑥 = ∅))
147	146	adantr 276	. . . . 5 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ (ω ⊔ 1o)) → (𝑥 = ∅ ∨ ¬ 𝑥 = ∅))
148	100, 144, 147	mpjaodan 799	. . . 4 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ (ω ⊔ 1o)) → (𝑥 = (𝐺‘𝑦) ↔ 𝑦 = if(𝑥 = ∅, (inr‘𝑥), (inl‘∪ 𝑥))))
149	148	adantl 277	. . 3 ⊢ ((⊤ ∧ (𝑥 ∈ ω ∧ 𝑦 ∈ (ω ⊔ 1o))) → (𝑥 = (𝐺‘𝑦) ↔ 𝑦 = if(𝑥 = ∅, (inr‘𝑥), (inl‘∪ 𝑥))))
150	1, 13, 32, 149	f1o2d 6099	. 2 ⊢ (⊤ → 𝐹:ω–1-1-onto→(ω ⊔ 1o))
151	150	mptru 1373	1 ⊢ 𝐹:ω–1-1-onto→(ω ⊔ 1o)

Syntax hints:  ¬ wn 3   ∧ wa 104   ↔ wb 105   ∨ wo 709  DECID wdc 835   = wceq 1364  ⊤wtru 1365   ∈ wcel 2160   ≠ wne 2360  ∃wrex 2469  Vcvv 2752   ⊆ wss 3144  ∅c0 3437  ifcif 3549  ∪ cuni 3824   ↦ cmpt 4079   I cid 4306  Oncon0 4381  suc csuc 4383  ωcom 4607   ↾ cres 4646  Fun wfun 5229   Fn wfn 5230  ⟶wf 5231  –1-1-onto→wf1o 5234  ‘cfv 5235  1_oc1o 6434   ⊔ cdju 7066  inlcinl 7074  inrcinr 7075  casecdjucase 7112

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-iord 4384  df-on 4386  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-1st 6165  df-2nd 6166  df-1o 6441  df-dju 7067  df-inl 7076  df-inr 7077  df-case 7113

This theorem is referenced by:  omp1eom  7124